The hundreds board hunt

In this activity, students colour a hundreds square to explore patterns in the 3 times table. They then discover that if they add the digits of any multiple of 3, the result is a number that is also divisible by 3. (For example: 54 is a multiple of 3, and 5 + 4 = 9, which is divisible by 3.)

For large numbers, they may need to repeatedly add the digits until they get a single digit, known as the digital root of that number. (For example, the digital root of 756 is 9 because 7 + 5 + 6 = 18 and 1 + 8 = 9, which is divisible by 3.)

Before the activity, discuss what multiples are. (You can use the definitions and examples on the student page.) A good introduction is to play a game of Yes or No, where the students decide whether given numbers are multiples of 5 or not. 

Talk about what you would expect to show up on the calculator if you entered ÷ 5 = and if was a multiple of 5. (The calculator would show a whole number.) Compare this to the result if wasn’t a multiple of 5. (The calculator would display a decimal fraction.) (See the Numeracy Development Project [NDP] activity, In and Out, page 36, Book 4: Teaching Number Knowledge.)

Nines and Threes (Book 6: Teaching Multiplication and Division, page 38) uses place value materials to help students understand why it is that the digit total of a multiple of 3 is also a multiple of 3.

Extend the activity by investigating other divisibility rules

For stage 7:

• multiples of 2 have 0 or an even number in the ones place
• multiples of 5 have 0 or 5 in the ones place
• for multiples of 9, if you keep adding the digits in the multiple until you get a single digit, it will be 9 (this is its digital root), for example, 693: 6 + 9 + 3 = 18, 1 + 8 = 9
• multiples of 10 have 0 in the ones place.

For stage 8:

• multiples of 4 are always even, and if you look at only the tens and the ones places, that number will be a multiple of 4; for example, 1 328 is a multiple of 4 because 28 is divisible by 4. This works because all digits in the hundreds place and above represent multiples of 100, which is itself divisible by 4
• multiples of 6 are always even and have a digital root that is divisible by 3 (that is, they are divisible by both 2 and 3)
• multiples of 8 are always even, and if you look at only the hundreds, tens, and ones places, that number will be a multiple of 8. For example, 7 624 is a multiple of 8 because 624 is divisible by 8. This works because all digits in the thousands place and above are multiples of 1 000, which is itself divisible by 8.

1.

a. Descriptions may vary. The pattern is a diagonal one, with two uncoloured diagonals after each coloured diagonal.

b. Practical activity

c. 3 + = = = = … (or 3 + + = = = = …) will give you all the multiples of 3. Alternatively, if you divide each of the coloured-in numbers by 3 on your calculator, the answer will be a whole number.

2.

a. 102. (102 ÷ 3 = 34. There is no remainder.)

b. 198. The last multiple must be one of the last 3 numbers on the square. 198 is divisible by 3 with no remainder, but 199 and 200 aren’t.

c. Practical activity

3.

Answers will vary. One simple rule (generalisation) is: a number is a multiple of 3 if all its digits add up to a number that is a multiple of 3. For example, for 147, 1 + 4 + 7 = 12. 12 is divisible by 3 (12 ÷ 3 = 4), so 147 is a multiple of 3 (147 ÷ 3 = 49).

Note: If your addition gives you a number with two or more digits and you are not sure if it is divisible by 3, add up the new digits: if the number is divisible by 3, your final answer must be 3, 6, or 9. For example, for 198: 1 + 9 + 8 = 18, 1 + 8 = 9.

4.

Numbers tried will vary. The rule and the note in question 3 still apply. For example, for 2 835: 2 + 8 + 3 + 5 = 18, 1 + 8 = 9; and for 9 864: 9 + 8 + 6 + 4 = 27, 2 + 7 = 9. On the calculator, all multiples of 3 will have whole-number answers when divided by 3.